Your original sentence about "dognapping" was very long.
Shorter examples are easier to work with.
- It is time that the law be enforced.
- It is time that the law was enforced.
- It is time that the government was honest in its dealings.
- It is time that the government become honest in its dealings.
When you want something about the world to change, you also usually wish that the world had been that way in the past.
||About the present
||About the past
||It is time that the house be made clean
||It is time that the house was clean
||I would like the house to stop being dirty right now
||I would like the dirtiness of the house to be a thing of the past
"It is time that the program ended" implies that we wished that the change happened earlier.
"It is time that the program end" does not show whether you wanted the change, or proposed event, to happen in the past, or if you only wanted the change to occur in the present.